Theorem mircgrextend 26947

Description: Link congruence over a pair of mirror points. cf tgcgrextend 26750. (Contributed by Thierry Arnoux, 4-Oct-2020.)

Hypotheses

Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirtrcgr.e	⊢ ∼ = (cgrG‘𝐺)
mirtrcgr.m	⊢ 𝑀 = (𝑆‘𝐵)
mirtrcgr.n	⊢ 𝑁 = (𝑆‘𝑌)
mirtrcgr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirtrcgr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mirtrcgr.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
mirtrcgr.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
mircgrextend.1	⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))

Assertion

Ref	Expression
mircgrextend	⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))

Proof of Theorem mircgrextend

Step	Hyp	Ref	Expression
1		mirval.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. 2 ⊢ − = (dist‘𝐺)
3		mirval.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		mirtrcgr.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		mirtrcgr.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
8		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
9		mirtrcgr.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐵)
10	1, 2, 3, 7, 8, 4, 6, 9, 5	mircl 26926	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
11		mirtrcgr.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
12		mirtrcgr.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
13		mirtrcgr.n	. . 3 ⊢ 𝑁 = (𝑆‘𝑌)
14	1, 2, 3, 7, 8, 4, 12, 13, 11	mircl 26926	. 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃)
15	1, 2, 3, 7, 8, 4, 6, 9, 5	mirbtwn 26923	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴))
16	1, 2, 3, 4, 10, 6, 5, 15	tgbtwncom 26753	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(𝑀‘𝐴)))
17	1, 2, 3, 7, 8, 4, 12, 13, 11	mirbtwn 26923	. . 3 ⊢ (𝜑 → 𝑌 ∈ ((𝑁‘𝑋)𝐼𝑋))
18	1, 2, 3, 4, 14, 12, 11, 17	tgbtwncom 26753	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼(𝑁‘𝑋)))
19		mircgrextend.1	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))
20	1, 2, 3, 4, 5, 6, 11, 12, 19	tgcgrcomlr 26745	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋))
21	1, 2, 3, 7, 8, 4, 6, 9, 5	mircgr 26922	. . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴))
22	1, 2, 3, 7, 8, 4, 12, 13, 11	mircgr 26922	. . 3 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋))
23	20, 21, 22	3eqtr4d 2788	. 2 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋)))
24	1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23	tgcgrextend 26750	1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  distcds 16897  TarskiGcstrkg 26693  Itvcitv 26699  LineGclng 26700  cgrGccgrg 26775  pInvGcmir 26917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-trkgc 26713  df-trkgb 26714  df-trkgcb 26715  df-trkg 26718  df-mir 26918

This theorem is referenced by:  mirtrcgr  26948